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What is an "ideal"?

Let W be a collection of sets.

W is an ideal

iff

(0) W is non-empty

(1) B,A:-W => AuB :- W

(2) B:-W and A c B => A:-W

W is an ideal

iff

(0) W is non-empty

(1) B,A:-W => AuB :- W

(2) B:-W and A c B => A:-W

What is a "s-ideal"?

Let W be a collection of sets.

W is a s-ideal

iff

(0) W is non-empty

(1) [ /\n:-|N A[n]:-W ] => U(n:-|N)_A[n] :- W

(2) B:-W and A c B => A:-W

W is a s-ideal

iff

(0) W is non-empty

(1) [ /\n:-|N A[n]:-W ] => U(n:-|N)_A[n] :- W

(2) B:-W and A c B => A:-W

Let W be a s-ring in X.

Let y : W -> [0,oo] be countably addive, y(O)=0.

Let N(y) = {BcX : \/A:-W BcA and y(A)=0}.

What can we conclude about N(y)?

Let y : W -> [0,oo] be countably addive, y(O)=0.

Let N(y) = {BcX : \/A:-W BcA and y(A)=0}.

What can we conclude about N(y)?

N(y) is a s-ideal

page 68 in 2nd measure

page 68 in 2nd measure

Let W be a s-ring in X.

Let y : W -> [0,oo] be countably addive, y(O)=0.

Let W^ = {EcX : \/A,B:-W AcEcB and y(B\A)=0}.

What can we conclude about W^ ?

Let y : W -> [0,oo] be countably addive, y(O)=0.

Let W^ = {EcX : \/A,B:-W AcEcB and y(B\A)=0}.

What can we conclude about W^ ?

W^ is a s-ring

page 69 in 2nd measure

page 69 in 2nd measure

Let W be a ring of sets. Let y:W->|R* be additive.

Prove that if A,B:-W and AcB and y(B\A):-|R then

[ y(B):-|R <=> ??? ]

Prove that if A,B:-W and AcB and y(B\A):-|R then

[ y(B):-|R <=> ??? ]

[ y(B):-|R <=> y(A):-|R ]

page 70 in 2nd measure

page 70 in 2nd measure

Let W be a ring of sets. Let y:W->|R* be additive.

Prove that if A,B:-W and AcB and y(B\A):-|R then

y(A):-|R <=> ???

Prove that if A,B:-W and AcB and y(B\A):-|R then

y(A):-|R <=> ???

y(A):-|R <=> y(B):-|R

page 70 in 2nd measure

page 70 in 2nd measure

Let x,y be real numbers.

Suppose that |x+y| <= |x+iy|.

What can we conclude?

Suppose that |x+y| <= |x+iy|.

What can we conclude?

x*y <= 0

Let x,y be real numbers.

Suppose that |x+iy| <= |x+y|.

What can we conclude?

Suppose that |x+iy| <= |x+y|.

What can we conclude?

x*y >= 0

Let W be a ring in X. Let y : W -> [0,oo] be additive.

Let A1,A2,B1,B2 be sets in W, and let E c X.

Suppose:

A1 c E c B1 and y(B1 \ A1) = 0,

A2 c E c B2 and y(B2 \ A1) = 0.

What can we conclude?

Let A1,A2,B1,B2 be sets in W, and let E c X.

Suppose:

A1 c E c B1 and y(B1 \ A1) = 0,

A2 c E c B2 and y(B2 \ A1) = 0.

What can we conclude?

y(A1) = y(B1) = y(A2) = y(B2)

page 70 in 2nd measure

page 70 in 2nd measure

Let W be a s-ring in X.

Let y : W -> [0,oo] countably additive, y(O)=0.

Let W^ = { EcX : \/A,B:-W AcEcB and y(B\A)=0 }.

Define a countably additive function on W^, that is an extension of y.

Prove that it is countably additive.

(In this SM-collection, this function will be denoted y^.)

Let y : W -> [0,oo] countably additive, y(O)=0.

Let W^ = { EcX : \/A,B:-W AcEcB and y(B\A)=0 }.

Define a countably additive function on W^, that is an extension of y.

Prove that it is countably additive.

(In this SM-collection, this function will be denoted y^.)

y^(E) = y(A) = y(B),

where A,B:-W and AcEcB and y(B\A)=0

page 71 in 2nd measure

where A,B:-W and AcEcB and y(B\A)=0

page 71 in 2nd measure